The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths $10$ cm and $20$ cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is $12$ cm. If the height of the pillar is $20$ cm, then the total area, in sq cm, of all six surfaces of the pillar is

We have a vertical pillar (like a column) whose cross-section is a trapezium. Think of it like a prism where the top and bottom are trapeziums (identical), the sides are rectangles, and the pillar stands vertically with height 20 cm.

Given information:

Parallel sides of trapezium: 10 cm and 20 cm

Non-parallel sides: equal length (let's call this x)

Height of trapezium: 12 cm

Height of pillar: 20 cm

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Since the non-parallel sides are equal, we need to find their length using the Pythagorean theorem.

When we drop a perpendicular from the shorter parallel side to the longer one, we create a right triangle.

The horizontal distance from where the perpendicular meets the longer side to the end of the longer side is:

$\tfrac{20 - 10}{2} = \tfrac{10}{2} = 5$ cm

Since the trapezium is symmetric (non-parallel sides are equal), the perpendicular falls exactly in the middle of the difference.

Now we have a right triangle with:

Height = 12 cm (given perpendicular distance)

Base = 5 cm (calculated above)

Hypotenuse = x cm (non-parallel side we want to find)

Using Pythagorean theorem:

$x^2 = 12^2 + 5^2 = 144 + 25 = 169$

$x = \sqrt{169} = 13$ cm

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Our pillar has 6 faces total:

2 trapezoidal faces (top and bottom)

4 rectangular faces (the sides)

The 4 rectangular faces have dimensions:

Two rectangles: 13 cm × 20 cm (from the non-parallel sides)

One rectangle: 10 cm × 20 cm (from the shorter parallel side)

One rectangle: 20 cm × 20 cm (from the longer parallel side)

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Area of 2 Trapezoidal Faces:

Formula for trapezium area: $\tfrac{1}{2} \times \text{height} \times (\text{sum of parallel sides})$

Area of one trapezium = $\tfrac{1}{2} \times 12 \times (10 + 20) = \tfrac{1}{2} \times 12 \times 30 = 180$ cm²

Area of 2 trapeziums = $2 \times 180 = 360$ cm²

Area of 4 Rectangular Faces:

Two rectangles (non-parallel sides): $2 \times (13 \times 20) = 2 \times 260 = 520$ cm²

One rectangle (shorter parallel side): $10 \times 20 = 200$ cm²

One rectangle (longer parallel side): $20 \times 20 = 400$ cm²

Total area of rectangles = $520 + 200 + 400 = 1120$ cm²

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Total surface area = Area of trapeziums + Area of rectangles

$= 360 + 1120 = 1480$ cm²

Therefore, the total area of all six surfaces is 1480 cm².

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When finding surface area of prisms:

1. We identify the cross-section (trapezium here)
2. We count all faces (2 identical cross-sections + rectangular sides)
3. We calculate each face area separately
4. We add them all up

The trickiest part is usually finding missing dimensions using geometry principles like the Pythagorean theorem!